Necessary Conditions for the Generic Global Rigidity of Frameworks on Surfaces

European Journal of Combinatorics

2014

a b s t r a c tWe provide a constructive characterisation of circuits in the simple (2, 2)-sparsity matroid. A circuit is a simple graph G = (V , E) with |E| = 2|V | − 1 where the number of edges induced by any X V is at most 2|X | − 2. Insisting on simplicity results in the Henneberg 2 operation being adequate only when the graph is sufficiently connected. Thus we introduce 3 different join operations to complete the characterisation. Extensions are discussed to when the sparsity matroid is connected and this is applied to the theory of frameworks on surfaces, to provide a conjectured characterisation of when frameworks on an infinite circular cylinder are generically globally rigid.

“…Following the submission of this paper, (1) ⇒ (2) has been confirmed in [11]. Theorem 5.4 shows the equivalence of (2) and (4).…”

Section: Global Rigiditysupporting

confidence: 63%

“…A characterisation of circuits in M * (2, 1) would be a step towards proving the analogue of Conjecture 5.7 for frameworks on a surface of revolution [17], such as a cone [11].…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

A constructive characterisation of circuits in the simple (2,2)-sparsity matroid

European Journal of Combinatorics

2014

“…In particular in the field of combinatorial optimization [9]. As particular motivation for us we seek a means to understand when a generic realisation of a graph on a surface is globally rigid (unique up to isometries), see [7,8] for details on the geometry of this problem. A related construction for circuits in the (2, 3)-sparse matroid [1] was a vital aspect of the characterisation of global rigidity in the plane [6] and we expect that the construction here, along with the construction for circuits in the simple (2, 2)-sparse matroid, will be crucial in establishing analogues for global rigidity on surfaces supporting either two (e.g., the cylinder) or one (e.g., the cone or torus) tangentially acting isometries.…”

Section: Introductionmentioning

confidence: 99%

“…A related construction for circuits in the (2, 3)-sparse matroid [1] was a vital aspect of the characterisation of global rigidity in the plane [6] and we expect that the construction here, along with the construction for circuits in the simple (2, 2)-sparse matroid, will be crucial in establishing analogues for global rigidity on surfaces supporting either two (e.g., the cylinder) or one (e.g., the cone or torus) tangentially acting isometries. See [11,Conjecture 5.7] and [7,Conjecture 9.1].…”

Section: Introductionmentioning

confidence: 99%

A constructive characterisation of circuits in the simple (2,1)‐sparse matroid

McCourt

Journal of Graph Theory

2018

A simple graph G=(V,E) is a (2, 1)‐circuit if |E|=2|V| and |E(H)|≤2|V(H)|−1 for every proper subgraph H of G. Motivated, in part, by ongoing work to understand unique realisations of graphs on surfaces, we derive a constructive characterisation of (2, 1)‐circuits. The characterisation uses the well‐known 1‐extension and X‐replacement operations as well as several summation moves to glue together (2, 1)‐circuits over small cutsets.

Rigid Cylindrical Frameworks with Two Coincident Points

Jackson

Kaszanitzky

Graphs and Combinatorics

2018

We develop a rigidity theory for frameworks in R 3 which have two coincident points but are otherwise generic and only infinitesimal motions which are tangential to a family of cylinders induced by the realisation are considered. We then apply our results to show that vertex splitting, under the additional assumption that the new edge is redundant, preserves the property of being generically globally rigid on families of concentric cylinders.